Formal Definition
Another way to describe the localization of a ring R at a subset S is via category theory. If R is a ring and S is a subset, consider all R-algebras A, so that, under the canonical homomorphism R → A, every element of S is mapped to a unit. These algebras are the objects of a category, with R-algebra homomorphisms as morphisms. Then, the localization of R at S is the initial object of this category.
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